Properties

Label 1224.1.cb.b
Level $1224$
Weight $1$
Character orbit 1224.cb
Analytic conductor $0.611$
Analytic rank $0$
Dimension $4$
Projective image $D_{12}$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1224.cb (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.610855575463\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{12}\)
Projective field Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} -\zeta_{12}^{4} q^{3} + \zeta_{12}^{2} q^{4} -\zeta_{12}^{5} q^{6} + \zeta_{12}^{3} q^{8} -\zeta_{12}^{2} q^{9} +O(q^{10})\) \( q + \zeta_{12} q^{2} -\zeta_{12}^{4} q^{3} + \zeta_{12}^{2} q^{4} -\zeta_{12}^{5} q^{6} + \zeta_{12}^{3} q^{8} -\zeta_{12}^{2} q^{9} + ( 1 + \zeta_{12}^{5} ) q^{11} + q^{12} + \zeta_{12}^{4} q^{16} + \zeta_{12} q^{17} -\zeta_{12}^{3} q^{18} -\zeta_{12}^{3} q^{19} + ( -1 + \zeta_{12} ) q^{22} + \zeta_{12} q^{24} -\zeta_{12} q^{25} - q^{27} + \zeta_{12}^{5} q^{32} + ( \zeta_{12}^{3} - \zeta_{12}^{4} ) q^{33} + \zeta_{12}^{2} q^{34} -\zeta_{12}^{4} q^{36} -\zeta_{12}^{4} q^{38} + ( \zeta_{12}^{3} - \zeta_{12}^{4} ) q^{41} + ( -1 - \zeta_{12}^{2} ) q^{43} + ( -\zeta_{12} + \zeta_{12}^{2} ) q^{44} + \zeta_{12}^{2} q^{48} + \zeta_{12}^{5} q^{49} -\zeta_{12}^{2} q^{50} -\zeta_{12}^{5} q^{51} -\zeta_{12} q^{54} -\zeta_{12} q^{57} + ( -1 + \zeta_{12}^{4} ) q^{59} - q^{64} + ( \zeta_{12}^{4} - \zeta_{12}^{5} ) q^{66} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{67} + \zeta_{12}^{3} q^{68} -\zeta_{12}^{5} q^{72} + ( \zeta_{12} + \zeta_{12}^{2} ) q^{73} + \zeta_{12}^{5} q^{75} -\zeta_{12}^{5} q^{76} + \zeta_{12}^{4} q^{81} + ( \zeta_{12}^{4} - \zeta_{12}^{5} ) q^{82} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{86} + ( -\zeta_{12}^{2} + \zeta_{12}^{3} ) q^{88} + \zeta_{12}^{3} q^{96} + ( -\zeta_{12}^{2} + \zeta_{12}^{3} ) q^{97} - q^{98} + ( \zeta_{12} - \zeta_{12}^{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} + 2q^{4} - 2q^{9} + O(q^{10}) \) \( 4q + 2q^{3} + 2q^{4} - 2q^{9} + 4q^{11} + 4q^{12} - 2q^{16} - 4q^{22} - 4q^{27} + 2q^{33} + 2q^{34} + 2q^{36} + 2q^{38} + 2q^{41} - 6q^{43} + 2q^{44} + 2q^{48} - 2q^{50} - 6q^{59} - 4q^{64} - 2q^{66} + 2q^{73} - 2q^{81} - 2q^{82} - 2q^{88} - 2q^{97} - 4q^{98} - 2q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times\).

\(n\) \(137\) \(613\) \(649\) \(919\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(-1\) \(\zeta_{12}^{3}\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
115.1
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i 0.500000 0.866025i 0.500000 + 0.866025i 0 0.866025 0.500000i 0 1.00000i −0.500000 0.866025i 0
259.1 −0.866025 0.500000i 0.500000 0.866025i 0.500000 + 0.866025i 0 −0.866025 + 0.500000i 0 1.00000i −0.500000 0.866025i 0
931.1 −0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 0.866025i 0 −0.866025 0.500000i 0 1.00000i −0.500000 + 0.866025i 0
1075.1 0.866025 0.500000i 0.500000 + 0.866025i 0.500000 0.866025i 0 0.866025 + 0.500000i 0 1.00000i −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
153.n even 12 1 inner
1224.cb odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1224.1.cb.b yes 4
3.b odd 2 1 3672.1.cc.a 4
8.d odd 2 1 CM 1224.1.cb.b yes 4
9.c even 3 1 1224.1.cb.a 4
9.d odd 6 1 3672.1.cc.b 4
17.c even 4 1 1224.1.cb.a 4
24.f even 2 1 3672.1.cc.a 4
51.f odd 4 1 3672.1.cc.b 4
72.l even 6 1 3672.1.cc.b 4
72.p odd 6 1 1224.1.cb.a 4
136.j odd 4 1 1224.1.cb.a 4
153.m odd 12 1 3672.1.cc.a 4
153.n even 12 1 inner 1224.1.cb.b yes 4
408.q even 4 1 3672.1.cc.b 4
1224.ca even 12 1 3672.1.cc.a 4
1224.cb odd 12 1 inner 1224.1.cb.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1224.1.cb.a 4 9.c even 3 1
1224.1.cb.a 4 17.c even 4 1
1224.1.cb.a 4 72.p odd 6 1
1224.1.cb.a 4 136.j odd 4 1
1224.1.cb.b yes 4 1.a even 1 1 trivial
1224.1.cb.b yes 4 8.d odd 2 1 CM
1224.1.cb.b yes 4 153.n even 12 1 inner
1224.1.cb.b yes 4 1224.cb odd 12 1 inner
3672.1.cc.a 4 3.b odd 2 1
3672.1.cc.a 4 24.f even 2 1
3672.1.cc.a 4 153.m odd 12 1
3672.1.cc.a 4 1224.ca even 12 1
3672.1.cc.b 4 9.d odd 6 1
3672.1.cc.b 4 51.f odd 4 1
3672.1.cc.b 4 72.l even 6 1
3672.1.cc.b 4 408.q even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{4} - 4 T_{11}^{3} + 5 T_{11}^{2} - 2 T_{11} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1224, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( ( 1 - T + T^{2} )^{2} \)
$5$ \( 1 - T^{4} + T^{8} \)
$7$ \( 1 - T^{4} + T^{8} \)
$11$ \( ( 1 - T )^{4}( 1 - T^{2} + T^{4} ) \)
$13$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
$17$ \( 1 - T^{2} + T^{4} \)
$19$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$23$ \( 1 - T^{4} + T^{8} \)
$29$ \( 1 - T^{4} + T^{8} \)
$31$ \( 1 - T^{4} + T^{8} \)
$37$ \( ( 1 + T^{4} )^{2} \)
$41$ \( ( 1 - T + T^{2} )^{2}( 1 + T^{2} )^{2} \)
$43$ \( ( 1 + T )^{4}( 1 + T + T^{2} )^{2} \)
$47$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
$53$ \( ( 1 + T^{2} )^{4} \)
$59$ \( ( 1 + T )^{4}( 1 + T + T^{2} )^{2} \)
$61$ \( 1 - T^{4} + T^{8} \)
$67$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
$71$ \( ( 1 + T^{4} )^{2} \)
$73$ \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
$79$ \( 1 - T^{4} + T^{8} \)
$83$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
$89$ \( ( 1 + T^{2} )^{4} \)
$97$ \( ( 1 + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
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